A generalization of plexes of Latin squares

A k-plex of a latin square is a collection of cells representing each row, column, and symbol precisely k times. The classic case of k = 1 is more commonly known as a transversal. We introduce the concept of a k-weight, an integral weight function on the cells of a latin square whose row, column, and symbol sums are all k. We then show that several non-existence results about k-plexes can been seen as more general facts about k-weights and that the weight-analogues of several well-known existence conjectures for plexes actually hold for k-weights.